Piero della Francesca

Piero della Francesca, who came from a family of fairly prosperous merchants, is recognised as one of the most important painters of the Renaissance. In his own time he was also known as a highly competent mathematician. In his Lives of the most famous painters ..., Giorgio Vasari (1511 - 1572) says that Piero showed mathematical ability in his earliest youth and went on to write 'many' mathematical treatises.

Of these, three are now known to survive. The titles by which they are known are: Abacus treatise (Trattato d'abaco), Short book on the five regular solids (Libellus de quinque corporibus regularibus) and On perspective for painting (De prospectiva pingendi). Piero almost certainly wrote all three works in the vernacular (his native dialect was Tuscan), and all three are in the style associated with the tradition of 'practical mathematics', that is, they consist largely of series of worked examples, with rather little discursive text.

The Abacus treatise is similar to works used for instructional purposes in 'Abacus schools'. It deals with arithmetic, starting with the use of fractions, and works through series of standard problems, then it turns to algebra, and works through similarly standard problems, then it turns to geometry and works through rather more problems than is standard before (without warning) coming up with some entirely original three-dimensional problems involving two of the 'Archimedean polyhedra' (those now known as the truncated tetrahedron and the cuboctahedron ).

Four more Archimedeans appear in the Short book on the five regular solids : the truncated cube, the truncated octahedron , the truncated icosahedron and the truncated dodecahedron . (All these modern names are due to Johannes Kepler (1619).) Piero appears to have been the independent re-discoverer of these six solids. Moreover, the way he describes their properties makes it clear that he has in fact invented the notion of truncation in its modern mathematical sense.

On perspective for painting is the first treatise to deal with the mathematics of perspective, a technique for giving an appearance of the third dimension in two-dimensional works such as paintings or sculptured reliefs. Piero is determined to show that this technique is firmly based on the science of vision (as it was understood in his time). He accordingly starts with a series of mathematical theorems, some taken from the optical work of Euclid (possibly through medieval sources) but some original to Piero himself. Some of these theorems are of independent mathematical interest, but on the whole the work is conceived as a manual for teaching painters to draw in perspective, and the detailed drawing instructions are mind-numbing in their repetitiousness. There are many diagrams and illustrations, but unfortunately none of the known manuscripts has illustrations actually drawn by Piero himself.

None of Piero's mathematical work was published under his own name in the Renaissance, but it seems to have circulated quite widely in manuscript and became influential through its incorporation into the works of others. Much of Piero's algebra appears in Pacioli 's Summa (1494), much of his work on the Archimedeans appears in Pacioli 's De divina proportione (1509), and the simpler parts of Piero's perspective treatise were incorporated into almost all subsequent treatises on perspective addressed to painters.

Source:School of Mathematics and Statistics University of St Andrews, Scotland